The standard reference for the statement that "any abstract variety is an open subscheme of a complete variety" is Nagata's 1962 paper Imbedding of an abstract variety in a complete variety. Unfortunately, this paper was apparently written before the language of schemes became standard, and uses Nagata's own language for "algebraic geometry over a Dedekind domain." Does anyone know of a translation of this proof (or another of the same statement) into scheme-theoretic language (or other language more comprehensible to the contemporary reader)?